(II) Go to (I) with \(i \leftarrow i+1\)  until

 Pi - Pi-1  < , 
for some very small positive constant \(\epsilon \)  and \(i \ge 1\) .

where the quasi-optimal control law is obtained by recursively solving learningeqx for \(K_{i+1}\)  without using the model of the plant. In learningeqx, the number of data sets \(l\)  is an integer that satisfies the condition \(l \ge n(n+1)/2 + mn\)  so that the equation is overdetermined.
When the plant is linear, the solution of learningeqx is guaranteed to converge to the LQR solution. For completeness, we summarize the results given in literature [1]} as Theorem 1.

Suppose Assumption 1 holds, and \(K_{i+1}\)  in learningeqx has a unique solution for all \(i = 0,1, \cdots \) . Then, the solution of the recursive equation learningeqx asymptotically converges to \(K^\star \)  as \(i \rightarrow \infty \) , where \(K^\star \)  is the LQR control law for the linear system \(\dot{{x}} = A {x} + B {u}\) .
